Proper orthogonal decomposition based seismic source wavefield reconstruction for finite element reverse time migration

The large storage requirement is a critical issue in cross-correlation imaging-condition based reverse time migration(RTM),because it requires the operation of the source and receiver wavefields at the same time.The boundary value method(BVM),based on the finite difference method(FDM),can be used to reconstruct the source wavefield in the reverse time propagation in the same way as the receiver wavefield,which can reduce the storage burden of the RTM data.Considering that the FDM cannot well handle models with discontinuous material properties and rough interfaces,we develop a source wavefield reconstruction strategy based on the finite element method(FEM),using proper orthogonal decomposition(POD)to enhance computational efficiency.In this method,we divide the whole time period into several segments,and construct the POD basis functions to get a reduced order model(ROM)for the source wavefield reconstruction in each segment.We show the corresponding quantitative analysis of the storage requirement of the POD-FEM.Numerical tests on the homogeneous model show the effectiveness of the proposed method,while the layered model and part of the Marmousi model tests indicate that the POD-FEM can keep an excellent balance between computational efficiency and memory usage compared with the full-stored method(FSM)and the BVM,and can be effectively applied in imaging.

Coherency matrix-based proper orthogonal decomposition with application to wind field simulation

Proper Orthogonal Decomposition （POD） provides a powerful modal transformation tool for stochastic dynamics. In this paper, coherency matrix-based proper orthogonal decomposition （CPOD） is presented as an innovative form of the POD based on cross power spectral density matrices. By introducing a discretizing scheme, the CPOD-based spectral representation method is obtained for use in stochastic simulation. Moreover, some criteria are proposed that allow the truncation order of CPOD to be conveniently determined. A numerical example to illustrate the application of the proposed method for the simulation of a wind velocity field is provided.

Coherent structures over riblets in turbulent boundary layer studied by combining time-resolved particle image velocimetry(TRPIV),proper orthogonal decomposition(POD),and finite-time Lyapunov exponent(FTLE)

Time-resolved particle image velocimetry（TRPIV） experiments are performed to investigate the coherent structure＇s performance of riblets in a turbulent boundary layer（TBL） at a friction Reynolds number of 185. To visualize the energetic large-scale coherent structures（CSs） over a smooth surface and riblets, the proper orthogonal decomposition（POD） and finite-time Lyapunov exponent（FTLE） are used to identify the CSs in the TBL. Spatial-temporal correlation is implemented to obtain the characters and transport properties of typical CSs in the FTLE fields. The results demonstrate that the generic flow structures, such as hairpin-like vortices, are also observed in the boundary layer flow over the riblets, consistent with its smooth counterpart. Low-order POD modes are more sensitive to the riblets in comparison with the high-order ones,and the wall-normal movement of the most energy-containing structures are suppressed over riblets. The spatial correlation analysis of the FTLE fields indicates that the evolution process of the hairpin vortex over riblets are inhibited. An apparent decrease of the convection velocity over riblets is noted, which is believed to reduce the ejection/sweep motions associated with high shear stress from the viscous sublayer. These reductions exhibit inhibition of momentum transfer among the structures near the wall in the TBL flows.

The proper orthogonal decomposition method for the Navier-Stokes equations

The proper orthogonal decomposition (POD) method for the instationary Navier-Stokes equations is considered. Several numerical approaches to evaluating the POD eigenfunctions are presented. The POD eigenfunctions are applied as a basis for a Galerkin projection of the instationary Navier-Stokes equations. And a low-dimensional ordinary differential models for fluid flows governed by the instationary Navier-Stokes equations are constructed. The numerical examples show that the method is feasible and efficient for optimal control of fluids.

Model reduction for supersonic cavity flow using proper orthogonal decomposition(POD)and Galerkin projection

The reduced-order model （ROM） for the two-dimensional supersonic cavity flow based on proper orthogonal decomposition （POD） and Galerkin projection is investigated. Presently, popular ROMs in cavity flows are based on an isentropic assumption, valid only for flows at low or moderate Mach numbers. A new ROM is constructed involving primitive variables of the fully compressible Navier-Stokes （N-S） equations, which is suitable for flows at high Mach numbers. Compared with the direct numerical simulation （DNS） results, the proposed model predicts flow dynamics （e.g., dominant frequency and amplitude） accurately for supersonic cavity flows, and is robust. The comparison between the present transient flow fields and those of the DNS shows that the proposed ROM can capture self-sustained oscillations of a shear layer. In addition, the present model reduction method can be easily extended to other supersonic flows.

Proper orthogonal decomposition analysis of coherent motions in a turbulent annular jet

A three-dimensional incompressible annular jet is simulated by the large eddy simulation(LES)method at a Reynolds number Re=8500.The time-averaged velocity field shows an asymmetric wake behind the central bluff-body although the flow geometry is symmetric.The proper orthogonal decomposition(POD)analysis of the velocity fluctuation vectors is conducted to study the flow dynamics of the wake flow.The distribution of turbulent kinetic energy across the three-dimensional POD modes shows that the first four eigenmodes each capture more than 1%of the turbulent kinetic energy,and hence their impact on the wake dynamics is studied.The results demonstrate that the asymmetric mean flow in the near-field of the annular jet is related to the first two POD modes which correspond to a radial shift of the stagnation point.The modes 3 and 4 involve the stretching or squeezing effects of the recirculation region in the radial direction.In addition,the spatial structure of these four POD eigenmodes also shows the counter-rotating vortices in the streamwise direction downstream of the flow reversal region.

Wind Pressures on Structures by Proper Orthogonal Decomposition

Proper orthogonal decomposition （POD） is an effective statistical technique for data reduction and feature extraction of the random field including the wind field. This paper introduces the theory of the POD and illustrates engineering of structures. Using the POD technique, it is shown that wind pressure data can be accurately reconstructed with a limited number of modes using the wind pressure data from wind tunnel test. Comparing the reconstructed values by POD with the original measured values from the wind tunnel test both in the time and frequency domains, it is concluded that the proper orthogonal decomposition（POD） is an efficient and practical technique for deriving the random wind pressure field from limited known data as shown in the pitched roof example in this paper.

On Orthogonal Decompositions of "Pull Back" Hermitian Holomorphic Vector Bundles

Let f ： Ω→Gr（n,H） be a holomorphic curve, where Ω is a bounded open simple connected domain on the complex plane C and Gr（n,H） the Grassmannian manifold. Denote by Ef the ＂pull back＂ bundle induced by f. We show the uniqueness of the orthogonal decomposition for those complex bundles. As a direct application, we give a complete description of the HIR decomposition of a Cowen- Douglas operator T ∈ Bn（Ω）. Moreover, we compute the maximal self-adjoint subalgebra of A＇（Ef） and A＇（T） respectively. Finally, we fix the masa of A＇（Ef） and .A＇ （T） which depends on the HIR decomposition of Ef or T respectively.

Review of Proper Orthogonal Decomposition Method Applied to Computational Aeroelasticity

Over the recent years there has been an increased trend in the use of Reduced Order Models （ROM） for modeling the coupled aeroelastic system. Of all the ROM models, the Proper Orthogonal Decomposition Method （POD） has been the most widely used, reason being the relative simplicity of implementation and the physical insight that it offers towards the physical problem. In this paper we begin by briefly recalling the recent work using POD for the computational aeroelasticity followed by the mathematical formulation. Mathematical formulation is important as it provides understanding of how POD method works. Implementation issues related to the POD method are presented next. Since POD is an empirical technique therefore, it is marred by the robustness issues as is the case with all the ROMs. In the end the variations of POD method, developed over the years are presented along with the most recent trend of using hybrid ROM.

Numerical investigation of the effect of wedge-type cavitating-bubble generator on attached unsteady cavitating flows using proper orthogonal decomposition method

The objective of this work is to reveal the effect of a passive control method called wedge-type cavitating-bubble generator(WCG)on the cloud cavitation dynamics of National Advisory Committee for Aeronautics(NACA)66 hydrofoil.The simulations are performed using the Partially-averaged Navier-Stokes(PANS)method coupled with the Zwart cavitation model.The proper orthogonal decomposition(POD)method is applied to extract the dominant flow structures.The results show that the WCG can induce the attached cavity to occur just behind the WCG instead of the hydrofoil leading edge.During the periodical time-evolution process of the unsteady cavity,it is found that the attached cavity with a larger scale around the hydrofoil with WCG has a rougher surface,accompanied with more shedding behaviors of small cavities.This is further illustrated by the POD modes,that is,the mode 1 and modes 2–4 present the large and small cavity vortex structures respectively.Meanwhile,the dominant frequencies of 50 Hz,47.5 Hz are given by the POD method respectively for the hydrofoils without and with WCG,which is in good agreement with that of FFT analysis.In addition,the correlation distribution of POD modal coefficients shows that the WCG can strengthen the vortex energy as well as the turbulence intensity.

Experimental Study and Simulation of Cavitation Shedding in Diesel Engine Nozzle using Proper Orthogonal Decomposition and Large Eddy Simulation

The unsteady cloud cavitation shedding in fuel nozzles greatly influences the flow characteristics and spray break-up of fuel,thereby causing erosion damage.With the application of high-pressure common rail systems in diesel engines,this phenomenon frequently occurs in the nozzle;however,cloud cavitation shedding frequency and its mechanism have yet to be studied in detail.In this study,a visualization experiment and proper orthogonal decomposition(POD)method were used to study the variations in the cavitation shedding frequency and analyze the cavitation flow structure in a 3 mm square nozzle.In addition,large eddy simulation(LES)was performed to explore the causes of cavitation shedding,and the relationship between cavitation and vortices.With the increase of the inlet and outlet pressure differences,and fuel temperatures,the degree of cavitation intensified and the frequency of cavitation cloud shedding gradually decreased.LES demonstrated the relationship between the vortices,and the development,shedding,and collapse of the cavitation clouds.Further,the re-entrant jet mechanism was found to be the main reason for the shedding of cavitation clouds.Through comparative experiments,the fluctuation of the vapor volume fraction in the nozzle hole accurately predicted the regions with stable cavitation,re-entrant jet,cavitation cloud shedding,and collapse.The frequency of cavitation shedding can then be calculated.This study employed an instantaneous POD method based on instantaneous cavitation images,which can distinguish the evolution process and characteristics of cavitation in the nozzle hole of diesel engines.

Finite difference scheme based on proper orthogonal decomposition for the nonstationary Navier-Stokes equations

The proper orthogonal decomposition(POD)and the singular value decomposition(SVD) are used to study the finite difference scheme(FDS)for the nonstationary Navier-Stokes equations. Ensembles of data are compiled from the transient solutions computed from the discrete equation system derived from the FDS for the nonstationary Navier-Stokes equations.The optimal orthogonal bases are reconstructed by the elements of the ensemble with POD and SVD.Combining the above procedures with a Galerkin projection approach yields a new optimizing FDS model with lower dimensions and a high accuracy for the nonstationary Navier-Stokes equations.The errors between POD approximate solutions and FDS solutions are analyzed.It is shown by considering the results obtained for numerical simulations of cavity flows that the error between POD approximate solution and FDS solution is consistent with theoretical results.Moreover,it is also shown that this validates the feasibility and efficiency of POD method.

Finite element formulation based on proper orthogonal decomposition for parabolic equations

A proper orthogonal decomposition (POD) method is applied to a usual finite element (FE) formulation for parabolic equations so that it is reduced into a POD FE formulation with lower dimensions and enough high accuracy. The errors between the reduced POD FE solution and the usual FE solution are analyzed. It is shown by numerical examples that the results of numerical computations are consistent with theoretical conclusions. Moreover, it is also shown that this validates the feasibility and efficiency of POD method.

Large eddy simulation and proper orthogonal decomposition analysis of turbulent flows in a direct injection spark ignition engine:Cyclic variation and effect of valve lift

Large eddy simulation(LES)is used to calculate the in-cylinder turbulent flow field in a direct injection spark ignition(DISI)engine.The computations are carried out for three different maximum valve lifts(MVL)and throughout 100 consecutive engine cycles.The simulated results as well as corresponding particle image velocimetry(PIV)measurement database are analyzed by the proper orthogonal decomposition(POD)method.Through a new developed POD quadruple decomposition the instantaneous in-cylinder flow fields are decomposed into four parts,named mean field,coherent field,transition field and turbulent field,respectively.Then the in-cylinder turbulent flow characteristics and cycle-to-cycle variations(CCV)are studied separately upon the four part flow fields.Results indicate that each part exhibits its specific characteristics and has close connection with others.The mean part contains more than 50%of the total kinetic energy and the energy cascade phenomenon occurs among the four part fields;the coherent field part possesses the highest CCV level which dominates CCV of the bulk flow.In addition,it is observed that a change in MVL affects significantly the in-cylinder flow behavior including CCV,especially for the coherent part.Furthermore,the POD analysis demonstrates that at least 25 sample cycles for the mean velocity and 50 sample cycles for the RMS velocity are necessary for obtaining converged and correct results in CCV.

Analysis of coherent structures in drag-reducing polymer solution flow based on proper orthogonal decomposition

Direct numerical simulation(DNS) of forcing homogeneous isotropic turbulence with polymers was performed.In order to understand the polymers effect on turbulent coherent structures,proper orthogonal decomposition was performed to identify coherent structures based on DNS data,so as to analyze the remarkable difference due to the addition of polymers.The results showed that the numbers for eigenmodes required for capturing coherent structures were 32 and 24 for the Newtonian fluid and polymer solution flows,respectively,which means the decrease of the complexity in polymer solution flow.Through the POD energy spectrum,it was found that the turbulent kinetic energy is distributed onto a large number of eigenmodes whether in the Newtonian fluid flow or polymer solution flow,suggesting that polymer solution flow is still turbulent in one aspect.Besides,the POD eigenmodes were investigated,which found that the small-scale structures are inhibited in polymer solution flow.

A High Robust Audio Watermarking Scheme Based on Orthogonal Decomposition

A high robust approach for the additive spread spectrum audio watermarking scheme based on orthogonal decomposition of the host signal has been proposed.By analyzing traditional spread spectrum（SS）,the section of the audio signal which is parallel to the direction of watermark signal acts as an interference source in decoder.The watermark is only embedded in the section which is perpendicular to watermark signal in the proposed scheme.In this way,the interference of host signal can be removed completely in decoder.Whether the watermarks have been embedded in the audio or not is researched and an optimal threshold in detector is given to achieve the minimum false alarm probability.The simulation results show that the proposed method can achieve high robustness against various attacks.

An Extraction of the Dominant Rotor-Stator Interaction Modes by the Use of Proper Orthogonal Decomposition (POD)

The Proper Orthogonal Decomposition method is applied to the instantaneous velocity field within the rotor-stator inter-row region of a high-speed high-pressure centrifugal compressor. The processed data come from experiments and numerical simulations. In comparison with a Fourier transform, the POD gives the best modal approximation for both initial fields, in terms of the energy expressed on any given number of modes to be taken into account： to reach 98% of the total energy of the velocity field, the required number of POD modes is around nine times smaller than the number of Fourier harmonics. The individual POD modes are given and show that the unsteady rotor-stator interaction is already present in the very first modes.

Knowledge Mining of Low Specific Speed Centrifugal Pump Impeller Based on Proper Orthogonal Decomposition Method

To clarify the complex relation between the pump blade shape and its corresponding hydraulic performance,the knowledge mining method of centrifugal pump impeller based on proper orthogonal decomposition(POD)was proposed.The pump blade shape was parameterized by cubic Bezier curve.The Latin hypercube design method was employed to supply the necessary samples for producing the perturbations of blade wrap angle,and blade angle at inlet and outlet.The hydraulic efficiency and head were optimized by NSGA-II and RBF hybrid algorithm.The Pareto-optimal solutions were obtained.In order to further illustrate the relationship between the centrifugal pump blade shape and its hydraulic performance,the POD method was used to discover the effects of optimized blade shape to the flow solutions.For the optimization of centrifugal pump MH48-12.5,blade shape and relative velocity field in impeller from Pareto-optimal solutions were analyzed.The results demonstrate that larger blade angle and smaller wrap angle increase the average kinetic energy in impeller,resulting in higher pump head design.Smaller blade angle and larger wrap angle decrease the velocity gradient from the pressure side to suction side,resulting in smaller hydraulic loss and higher efficiency design.

Orthogonal Decomposition of Incomplete-Profile Finite Game Space

This work studies the orthogonal decomposition of the incomplete-profile normal finite game(IPNFG)space using the method of semi-tensor product(STP)of matrices.Firstly,by calculating the rank of the incomplete-profile potential matrix,the bases of incomplete-profile potential game subspace(GPΩ)and incomplete-profile non-strategic game subspace(NΩ)are obtained.Then the bases of incomplete-profile pure potential game subspace(PΩ)and incomplete-profile pure harmonic game subspace(HΩ)are also revealed.These bases offer an expression for the orthogonal decomposition.Finally,an example is provided to verify the theoretical results.

FAST IMPLEMENTATION OF ORTHOGONAL EMPIRICAL MODE DECOMPOSITION AND ITS APPLIGA-TION INTO HARMONIC DETECTION

Since the empirical mode decomposition （EMD） lacks strict orthogonality, the method of orthogonal empirical mode decomposition （OEMD） is innovationally proposed. The primary thought of this method is to obtain the intrinsic mode function （IMF） and the residual function by auto-adaptive band-pass filtering. OEMD is proved to preserve strict orthogonality and completeness theoretically, and the orthogonal basis function of OEMD is generated, then an algorithm to implement OEMD fast, IMF binary searching algorithm is built based on the point that the analytical band-pass filtering preserves perfect band-pass feature in the frequency domain. The application into harmonic detection shows that OEMD successfully conquers mode aliasing, avoids the occurrence of false mode, and is featured by fast computing speed. Furthermore, it can achieve harmonic detection accurately combined with the least square method.